Rapid Recall
- Recall for a two dimensional region that $\bar x = \dfrac{\iint_R xdA}{\iint_R dA}$. If you know that the area of a region is $A = 3$ and the centroid is $(5,7)$, compute $\iint_R 2xdA$.
Solution
Since $\bar x = \dfrac{\iint_R xdA}{\iint_R dA}$, we know that $\iint_R xdA = \bar x \iint_R dA = \bar x A$. In other words, we can replace $\iint_R xdA$ with $\bar x A$. We then compute $$\iint_R 2xdA = 2\iint_R xdA = 2\bar x A = 2(5)(3) = 30.$$
- A curve $C$ traverses around a region $R$ with $A = 3$ and the centroid is $(5,7)$. Compute the work done by $\vec F = (2x+3y,x^2+y^2)$ along $C$ (Recall Green's theorem: $\oint Mdx+Ndy = \iint_R N_x-M_y dA$).
Solution
Green's theorem will apply here, because we have a simple closed curve. This gives $$\iint_R N_x-M_y dA =\iint_R 2x-3 dA =2\iint_R xdA - \iint_R 3 dA = 2\bar xA - 3A = 2(5)(3)-3(3) = 21.$$
- For the surface $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$, compute the normal vector $\vec n = \frac{\partial \vec r}{\partial u}\times \frac{\partial \vec r}{\partial v}$.
Solution
The partial derivatives are $$\frac{\partial \vec r}{\partial u} = (\cos v,\sin v,-2u),\quad \frac{\partial \vec r}{\partial v} = (-u\sin v,u\cos v,0), $$ The normal vector $\vec n$ is the cross product of these two, so $$\vec n = (2u^2\cos v, 2u\sin v,u\cos^2v+u\sin^2v) = (2u^2\cos v, 2u^2\sin v,u).$$
Group problems
- Compute the work done by $\vec F = (-3y,3x)$ to move an object counterclockwise once along the circle $\vec r(t) = (5\cos t, 5\sin t),$ using Green's theorem $\iint_R N_x-M_y dA$.
- Compute the work done by $\vec F = (y^2,3x)$ to move an object counterclockwise once along the circle $\vec r(t) = (5\cos t, 5\sin t),$ using Green's theorem $\iint_R N_x-M_y dA$.
- Compute the work done by each vector field below to move an object counterclockwise once along the triangle with corners $(0,0)$, $(2,0)$, and $(0,3)$.
- $\vec F = (2x-y,2x+4y)$
- $\vec F = (x^2+y^2,x+y)$
- Consider the parametric surface $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $u\in [1,2]$ and $v\in [0,2\pi]$.
- Compute a normal vector to the surface (so $\vec n = \vec r_u\times \vec r_v$).
- Give an equation of the tangent plane to the surface at $(u,v)=(3/2,\pi/2)$.
- Set up an integral to compute the surface area of the surface.
- Consider the surface parametrized by $\vec r(u,v) = (u, v, u^2+v^2)$ for $-3\leq u\leq 3$ and $0\leq v\leq 3$.
- Compute $d\sigma = \left |\dfrac{\partial \vec r}{\partial u}\times\dfrac{\partial \vec r}{\partial u}\right|dudv$.
- Set up an integral formula to compute $\bar z$ for this surface.
- Consider the parametric surface $\vec r(u,v) = (u^2\cos v, u, u^2\sin v)$ for $u\in [1,2]$ and $v\in [0,2\pi]$.
- Compute a normal vector to the surface (so $\vec n = \vec r_u\times \vec r_v$).
- Give an equation of the tangent plane to the surface at $(u,v)=(3/2,\pi/2)$.
- Set up an integral to compute the surface area of the surface.
- Draw each curve or surface given below.
- $\vec r(u,v) = (4\cos u,v, 3\sin u)$ for $0\leq u\leq \pi$ and $0\leq v\leq 7$.
- $\vec r(t) = (3\cos t,3\sin t,t)$ for $0\leq t\leq 6\pi$.
- $\vec r(u,v) = (u\cos v,u\sin v,v)$ for $0\leq v\leq 6\pi$ and $2\leq u\leq 4$.
- $\vec r(t) = (0,t,9-t^2)$ for $0\leq t\leq 3$.
- $\vec r(x,y) = (x,y,9-x^2-y^2)$ for for $0\leq x \leq 3$ and $-3\leq y\leq 3$.
- $\vec r(u,v) = (u\cos v,u\sin v,9-u^2)$ for $0\leq u\leq 3$ and $0\leq v\leq 2\pi$.
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